“Perfect obstruction theory”的意思、由来-开放百科全书

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Perfect obstruction theory

释义

Examples
Schemes Example 1 Deligne–Mumford stacks
Symmetric obstruction theory
Notes
References
See also

In algebraic geometry, given a Deligne–Mumford stack X, a perfect obstruction theory for X consists of:

a perfect two-term complex in the derived category of quasi-coherent étale sheaves on X, and
a morphism , where is the cotangent complex of X, that induces an isomorphism on and an epimorphism on .

The notion was introduced by {{harv|Behrend–Fantechi|1997}} for an application to the intersection theory on moduli stacks; in particular, to define a virtual fundamental class.

Examples

Schemes

Consider a regular embedding fitting into a cartesian square

where are smooth. Then, the complex

(in degrees )

forms a perfect obstruction theory for X.^[1] The map comes from the composition

This is a perfect obstruction theory because the complex comes equipped with a map to coming from the maps and . Note that the associated virtual fundamental class is

Example 1

Consider a smooth projective variety . If we set , then the perfect obstruction theory in is

and the associated virtual fundamental class is

In particular, if is a smooth local complete intersection then the perfect obstruction theory is the cotangent complex (which is the same as the truncated cotangent complex).

Deligne–Mumford stacks

The previous construction works too with Deligne–Mumford stacks.

Symmetric obstruction theory

By definition, a symmetric obstruction theory is a perfect obstruction theory together with nondegenerate symmetric bilinear form.

Example: Let f be a regular function on a smooth variety (or stack). Then the set of critical points of f carries a symmetric obstruction theory in a canonical way.

Example: Let M be a complex symplectic manifold. Then the (scheme-theoretic) intersection of Lagrangian submanifolds of M carries a canonical symmetric obstruction theory.

Notes

1. ^{{harvnb|Behrend–Fantechi|1997|loc=§ 6}}

References

{{Cite arXiv|last=Behrend|first=K.|eprint=math/0507523v2|title=Donaldson–Thomas invariants via microlocal geometry|year=2005}}
{{Cite journal|last=Behrend|first=K.|last2=Fantechi|first2=B.|date=1997-03-01|title=The intrinsic normal cone|journal=Inventiones Mathematicae|language=en|volume=128|issue=1|pages=45–88|doi=10.1007/s002220050136|issn=0020-9910|bibcode=1997InMat.128...45B|arxiv=alg-geom/9601010}}
{{Cite web|url=https://mathoverflow.net/questions/206804/understanding-the-obstruction-cone-of-a-symmetric-obstruction-theory/211932#211932|title=Understanding the obstruction cone of a symmetric obstruction theory|last=Oesinghaus|first=Jakob |website=MathOverflow|date=2015-07-20|access-date=2017-07-19}}